Theorem vneulem1 1131

Description: Part of von Neumann's lemma. Lemma 9, Kalmbach p. 96. (Contributed by NM, 15-Mar-2010.) (Revised by NM, 31-Mar-2011.)

Assertion

Ref	Expression
vneulem1	(((x ∪ y) ∪ u) ∩ w) = (((x ∪ y) ∪ u) ∩ ((u ∪ w) ∩ w))

Proof of Theorem vneulem1

Step	Hyp	Ref	Expression
1		leor 159	. . . 4 w ≤ (u ∪ w)
2		leid 148	. . . 4 w ≤ w
3	1, 2	ler2an 173	. . 3 w ≤ ((u ∪ w) ∩ w)
4		lear 161	. . 3 ((u ∪ w) ∩ w) ≤ w
5	3, 4	lebi 145	. 2 w = ((u ∪ w) ∩ w)
6	5	lan 77	1 (((x ∪ y) ∪ u) ∩ w) = (((x ∪ y) ∪ u) ∩ ((u ∪ w) ∩ w))

This proof depends on syntax axioms:   = wb 1   ∪ wo 6   ∩ wa 7

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is used by:  vneulem4  1134